Question

This is for AP CALCULUS. I really need help on this. I’m trying to find the first and second derivative of these three functions.

Answer 1

Review Material:

$\sqrt[n]{x^m} = x^{\frac{m}{n}}$

$\frac{d}{dx}[x^n]=nx^{n-1}$

$\frac{d}{dx}[\sin(x)]=\cos(x)$

 $\frac{d}{dx}[\cos(x)]=-\sin(x)$

$\frac{d}{dx}[constant]=0$

Step-by-step explanation:

(a)

$F(x) = -2\cos(x) +x^{\frac{4}{3}} -3e\\ \text{Note: 3e is a constant}\\F'(x) = 2\sin(x) + \frac{4}{3}x^{\frac{1}{3}}\\F''(x) = 2\cos(x) + \frac{4}{9}x^{-\frac{2}{3}}$

(b)

$F(x) = x^{-3} + \frac{1}{2}x^2-\sin(x)\\F'(x)=-3x^{-4} + x - \cos(x)\\F''(x) = 12x^{-5} + 1 + \sin(x)$

(c)

$F(x) = 2x^{-3}+x^{\frac{3}{4}}-4x\\F'(x) = -6x^{-4} + \frac{3}{4}x^{-\frac{1}{4}}-4\\F''(x)= 24x^{-5} - \frac{3}{16}x^{-\frac{5}{4}}$